Arbitrarily Long Increasing and Decreasing Runs in ⌊n^(p_n/n)⌋

Abstract

Let \(p_n\) denote the \(n\)th prime and define\[A_n=\left\lfloor n^{p_n/n}\right\rfloor.\]OEIS A318199 records the conjecture that no consecutive strictly increasing run contains more than 17 terms. Introducing \(B_n=n^{p_n/n}\) and \(g_n=p_{n+1}-p_n\), we derive the exact identity\[\log\frac{B_{n+1}}{B_n}=\frac{\log(n+1)}{n+1}(g_n-T_n),\qquadT_n=p_n\left(\frac{(n+1)\log n}{n\log(n+1)}-1\right),\]with \(T_n\sim\log p_n\). A quantitative margin transfers fixed proportional deviations of \(g_n\) from \(\log p_n\) to strict inequalities after taking floors. Ford–Maynard–Tao chains of consecutive large prime gaps then yield, for every fixed \(L\ge2\), infinitely many strictly increasing consecutive blocks of at least \(L\) terms. Maynard’s bounded-diameter prime clusters yield infinitely many strictly decreasing blocks of at least \(L\) terms. Hence both increasing and decreasing run lengths are unbounded. We also record an explicit 18-term increasing block beginning at \(n=10{,}073{,}436\), giving a finite counterexample to the stated bound of 17; its verification is separated from the analytic unboundedness proof and placed in the appendix.